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Optimal Sketching Bounds for Sparse Linear Regression
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:11288-11316, 2023.
Abstract
We study oblivious sketching for k-sparse linear regression under various loss functions. In particular, we are interested in a distribution over sketching matrices S∈Rm×n that does not depend on the inputs A∈Rn×d and b∈Rn, such that, given access to SA and Sb, we can recover a k-sparse ˜x∈Rd with ‖. Here \|\cdot\|_f: \mathbb R^n \rightarrow \mathbb R is some loss function – such as an \ell_p norm, or from a broad class of hinge-like loss functions, which includes the logistic and ReLU losses. We show that for sparse \ell_2 norm regression, there is a distribution over oblivious sketches with m=\Theta(k\log(d/k)/\varepsilon^2) rows, which is tight up to a constant factor. This extends to \ell_p loss with an additional additive O(k\log(k/\varepsilon)/\varepsilon^2) term in the upper bound. This establishes a surprising separation from the related sparse recovery problem, which is an important special case of sparse regression, where A is the identity matrix. For this problem, under the \ell_2 norm, we observe an upper bound of m=O(k \log (d)/\varepsilon + k\log(k/\varepsilon)/\varepsilon^2), showing that sparse recovery is strictly easier to sketch than sparse regression. For sparse regression under hinge-like loss functions including sparse logistic and sparse ReLU regression, we give the first known sketching bounds that achieve m = o(d) showing that m=O(\mu^2 k\log(\mu n d/\varepsilon)/\varepsilon^2) rows suffice, where \mu is a natural complexity parameter needed to obtain relative error bounds for these loss functions. We again show that this dimension is tight, up to lower order terms and the dependence on \mu. Finally, we show that similar sketching bounds can be achieved for LASSO regression, a popular convex relaxation of sparse regression, where one aims to minimize \|Ax-b\|_2^2+\lambda\|x\|_1 over x\in\mathbb{R}^d. We show that sketching dimension m =O(\log(d)/(\lambda \varepsilon)^2) suffices and that the dependence on d and \lambda is tight.